What will happen if you try to update record in After Trigger Context in Salesforce?

Typically, if you are populating trigger record fields, it is best to use a before trigger, because it does not add transaction cost. However, if you absolutely must populate information to trigger records in an after context, you can re-query for them and act on this new reference, then perform an update. This strategy increases the cost in both DML Statements and SOQL.

you can perform DML in trigger from future methods with callout=true.you can re-query for them and act on this new reference, then perform an update.

If you did this, it would fail because you're essentially trying to make a duplicate record here. You are correct, however, in that the After context has access to the ID(s) of the updated record(s). In fact, with After Update we have use of trigger.new, which contains not only the ID(s) but all the fields for the updated records. To make changes to any of these fields, you could simply loop through trigger.new, updating the field values if they need to be updated. This is important and is one of the many reasons why it's not best practice to perform updates to updated records in After Update - if you don't validate those field values and instead set up the loop to always change them and update all the records again, you will end up in a potentially infinite loop.
Long story short, don't do this. Use Before Update unless it's absolutely necessary.